a If s n is increasing and unbounded then s n b If s n is decreasing and

A if s n is increasing and unbounded then s n b if s

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(a)If{sn}is increasing and unbounded, thensn+.(b)If{sn}is decreasing and unbounded, thensn→ -∞. Proof: Cauchy Sequences Definition 5. A sequence { s n } is a Cauchy sequence if to each > 0 there is a positive integer N such that m, n > N implies | s n - s m | < . THEOREM 10. Every convergent sequence is a Cauchy sequence. Proof: Suppose s n s . Let > 0. There exists a positive integer N such that | s - s n | < / 2 for all n > N . Let n, m > N . Then | s m - s n | = | s m - s + s - s n | ≤ | s m - s | + | s - s n | < 2 + 2 = . Therefore { s n } is a Cauchy sequence. THEOREM 11. Every Cauchy sequence is bounded. 19
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Proof: Let { s n } be a Cauchy sequence. There exists a positive integer N such that | s n - s m | < 1 whenever n, m > M . Therefore | s n | = | s n - s N +1 + s N +1 | ≤ | s n - s N +1 | + | s N +1 | < 1 + | s N +1 | for all n > N. Now let M = max {| s 1 | , | s 2 | , . . ., | s N | , 1 + | s N +1 |} . Then | s n | ≤ M for all n . THEOREM 12.A sequence{sn}is convergent if and only if it is a Cauchy sequence. Exercises 2.3 1. True – False.Justify your answer by citing a theorem, giving a proof, or giving a counter-example.(a) If a monotone sequence is bounded, then it is convergent.(b) If a bounded sequence is monotone, then it is convergent.(c) If a convergent sequence is monotone, then it is bounded.(d) If a convergent sequence is bounded, then it is monotone.2. Give an example of a sequence having the given properties.(a) Cauchy, but not monotone.(b) Monotone, but not Cauchy.(c) Bounded, but not Cauchy.3. Show that the sequence{sn}defined bys1= 1andsn+1=14(sn+ 5)is monotone andbounded. Find the limit.4. Show that the sequence{sn}defined bys1= 2andsn+1=2sn+ 1is monotone andbounded. Find the limit.5. Show that the sequence{sn}defined bys1= 1andsn+1=sn+ 6is monotone andbounded. Find the limit.6. Prove that a bounded decreasing sequence converges to its greatest lower bound.7. Prove Theorem 9 (b).II.4.SUBSEQUENCESDefinition 6.Given a sequence{sn}.Let{nk}be a sequence of positive integers such thatn1< n2< n3<· · ·. The sequence{snk}is called asubsequenceof{sn}.ExamplesTHEOREM 13.If{sn}converges tos,then every subsequence{snkof{sn}also convergestos. 20
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CorollaryIf{sn}has a subsequence{tn}that converges toαand a subsequence{un}thatconverges toβwithα=β,then{sn}does not converge. Every bounded sequence has a convergent subsequence.
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